[Music] is a statement of nons satiation we can call it local non satiation another way to talk about a person is not satisfied for what he has we can bring let us put it one and we can bring what we call his monotony City or in other word in simple English we can say more is better so here you have again let me draw here you have a bundle X not here in this area in the SED area at least you have more of X1 or more of x2 in fact at most places more of X1 and X2 both so what we say all the bundles in the saided region would be preferred to X notot this is more is better or monotonicity is satisfied let us understand the relationship between the two nons satiation local nons satiation satiation and monotonicity can you see any relation between them if monotonicity property is satisfied local non satiation property is satisfied but not the other way around let us take an example here is a bundle so you take a small neighborhood no matter how small neighborhood you take you will find many such bundle which would be preferred to the original bundle so what local on satiation says that in in any small neighborhood that you take you will always find a bundle that is prefer to the original bundle so here you are able to find because of monotonicity but it is also very much possible that uh local nons satiation doesn't imply monotonicity it so happens that you always find a bundle in the opposite direction like let me say there is a person who says less is better what he does to compare X and Y what he does he calculate the total amount of both the good and let us go back to the two good world and see y1 + Y2 and he says if X1 + X2 is smaller than or equal to y1 + y Y 2 he prefers X or X is at least as good as Y what it means that you are moving in this direction so what you can do you can draw a line and see these things let me write here you know because x + X1 + X2 if wherever it is constant X1 + X2 this person is going to be indifferent everywhere where X1 + X2 is constant he is indifferent is indifferent and as you move towards origin your liking is increasing because you prefer less to more and everywhere local non satiation is satisfied except at one point you take a point here take a small neighborhood you can always move towards origin and you will have bundle that you like more the only place it is not satisfied once you hit the origin no matter how small neighborhood you take you take this neighborhood but the problem here is that negatives are not allowed these threes are not allowed so you are struck with this neighborhood and you do not have any more space to move in this direction to move in this direction and therefore local nons satiation is not satisfied at origin but everywhere else it is satisfied but monotonicity is nowhere satisfied so these could be two different statement again we are less particular about psychological assumption whether local nons satiation is satisfied or monotonicity is satisfied doesn't matter as long as the previous assumptions are satisfied we will get utility function but it just makes our life easier when we also talk about nons satiation and most of the time monoton onity is really applicable as we all prefer more to better but let me give you another example that when you know you have let us say fish and coconut and it so happens for the Robinson cruso stuck on that island that he cannot consume you know his he wants at most two units of fish and two units of C coconut and more if he takes more it is harmful for him it it causes diarrhea or lose motion to him so he wants this is his ideal position and he is he wants to be the closest to his ideal position so further he moves from his ideal position worse off he becomes so in this case the indifference map would be concentric Circle centered at 2 comma 2 of course my circles are not perfect but this is the way it would be and so all these let us say this is 10 this is 20 this is 30 and this is the kind of the map that you will get so U of f comma C can be written as let us say minus of f - 2 sare minus or we can do here plus you know C Min - 2 whole s² so that's why in this case of course 10 2030 will not work in this case highest utility possibly zero and at that this person gets at 2 comma 2 and everywhere else because if it is not 2 comma 2 then either this is a positive number or this is a positive number so utility is everywhere going to be negative number but that is much easier to deal with and deliberately I gave you this example so that you understand what we can do we can do monotonic transformation and everywhere if we add 100 then this the label will change Zero will become 100 10 - 10 will become 90 and this will become 80 and 70 and so on still you know in difference map would remain the same but the number written to a particular curve would become different because we have done monotonic transformation okay so we can say the map the shape of the map Contours of the map doesn't depend on the exact number but it it you get the same map whether you use one utility function or you take the monotonic transformation of that utility function so it should be clear to you so this we already did the graphical representation so now let us see uh now let us see here in the graphical representation what we have so once you know we talked about this kind of indifference curve here is X2 here is X1 and it is associated with 10 what does it mean that all the bundles on the curve gives the same level of utility 10 now once local or let us say nons satiation is satisfied satiation is satisfied to be more specific monotonicity is satisfied what does it mean that if you move in and I always make mistake this is perhaps uh Northeast Direction I think in one of the previous videos I said Northwest Direction I have very poor directional sense so you check for yourself direction is right I think this is Northeast Direction and in North East Direction once you move utility has to increase as it has more of X1 comma X2 so indifference curve what it means that an indifference curve should not be moving in North East Direction because you compare a bundle here and you compare a bundle here here let us say this is x and y y is greater than x all components be why because y1 is greater than X1 and Y2 is greater than X2 therefore we can say Y is greater than x so y has more of good one more of good two so y will get higher give higher utility y would be more preferred to X and therefore X and Y cannot be on the same indifference curve again remember what does an indifference curve mean indifference curve is a curve that passes through all those bundles which give the same level of Happiness same level of satisfaction same level of utility to the consumer so X and Y if monotonicity is satisfied X and Y cannot give the same level of satisfaction why will give higher level of satisfaction and therefore an indifference curve cannot be upward sloping and therefore this zone is not possible and let us look at it you know non even non satiation is not satisfy you take a small neighborhood here it's completely contained within this SED Zone which gives the same level of utility so even local non satiation is violated if local nons satiation is violated monity will definitely be violated but if monotonicity is violated it may possible that local nons satiation is not violated so we have to keep keep in mind so you will not have any thick Zone on indifference curve on any indifference curve and you will no thick Jone and second you will not have an upward sloping indifference curve so what we can further look at it so you have here it cannot in difference curve cannot go in this Direction cannot go in this Direction only possibility that's left is that indifference curve has to be a downward sloping like this and it makes sense it makes sense let us look at the downward sloping part here compare now let us say you have X and here you have have Y X and Y so in X and Y are on the same indifference curve it means you are getting the same level of utility and for example I'm just saying it's 10 clearly we have X1 is less than y1 and X2 is greater than Y2 so what is happening lack of x X1 is getting compensated by X2 so some compensation is happening because if you take one of the items you have this indifference curve and if you remove one of the items because of monotonicity you will have less of a satisfaction less of a utility then you need to increase the other item by some amount to bring that person back to same level of utility this is the decrease and this is the increase that we are having good so let us understand indifference curve what we have figured out that one we have figured out that indifference curve should be should be downward sloping thanks to monotonicity should not have should not have any thick Zone thank even monotonicity is applicable but monotonicity is stronger requirement even non local satiation or local non satiation will give you the same thing one more thing that we should understand that two indifference curve should not cross one another let us consider a scenario in which let us say this is Axis one for good one this is Axis two for good two let us take here a bundle a here at intersection is bundle B and here is C clearly C is greater than a as C2 is greater than A2 and C1 is greater than A1 therefore C is greater than a and monotonicity will say that c is preferred to a but what Crossing will ensure because this person is indifferent between A and B so we can write it like this a is at least as good as B and B is at least as good as a this is same as saying you are indifferent between a and b and you are indifferent between B and C what it means that b is at least as good as C and C is at least as good as B if you take if you take these two together from here you will get a is at least as good as C and if you take these two together you will get C is at least as good as a so if you take now these two what you are going to get is that you are indifferent between a and C and anyway here is 10 let us say this is also 10 that also says because a gives you utility 10 C gives you utility 10 so you are indifferent between a and C without doing all these things but anyway I wanted to illustrate uh the way of doing things methodically but we also figured out because of monotonicity C is preferred to a so it this contradicts that you are indifferent between a and C so what it means that this particular possibility is wrong so what we have figured out that not only indifference curve should be downward sloping should not have any thick Zone and know third is that no two indifference curve a curve should inter should intersect this is third of course here we are using transitivity also transitivity and [Music] monotonicity so that's what we have learned now we will move to a notion that we have learn a similar notion we have learned in a different context you are familiar with marginal rate of technical substitution that we did in case of isoquant the similar stuff we are going to do with indifference curve by now you must have understood that indifference curve and an isoquant graphically they look very similar so what is marginal rate of substitution you have let us say a downward sloping curve let us say axis one gives you amount of good one and axis to gives you amount of good to and marginal rate of technical substitution is basically slope of this indifference curve this is the slope what is the slope gives the slope gives gives your mental exchange rate so this mental exchange rate how many unit of um X2 you need to compensate you if you give up one unit of good one that's what would be the marginal rate of substitution the names are slightly different so you don't get confused marginal rate of substitution mrss is in case of consumer and mrts is in case of producer or manager that's what is important and so here also if we want to if we fix you know we say that why you know all the bundles on this curve gives the same level of utility so what we are having if we take any bundle X1 comma X2 let us say it GES gives you 10 for example so again we can have the same logic X1 X2 you are not free to vary on your own once you decide X1 to have the same level of utility you will have to decide X2 accordingly and from here again you do differentiate both side with respect to X1 that's what we did last time here also we can do we can do total differentiation I I'm deliberately using SL ly different technique so you can be written as U1 X1 let me write U1 so it means the derivative of just for mathematical understanding I'm going to use slightly different technique this is the technique I had used when we had done the mrts for ISO uh isoquant here we are using the total difference iation so we can do this can be written as d u d X1 it means we are partially differentiating U with respect to X1 keeping X2 fixed dx1 plus du dx2 dx2 is equal to Zer and from here again we get dx2 / dx1 is nothing but mu1 divided by mu2 m UI is D DX I that's what it is and so we get exactly the same formulation if you remember in case of mrts we had obtained dx2 by dx1 is equal to minus mp1 / mp2 and when we has used Liber as the first input and capital as the second input we had obtained MPL divided by mpk exactly the same thing you get so this is the marginal rate of substitution that you obtain here in this case now let us come to the last psychological assumption that we are going to make and that is convexity of preferences what is convexity of preferences so what we have figured out that indifference curve should not be thick it should be down Ward cross downward sloping and two indifference curves should not intersect one another but when it comes to downward sloping we can have three different ways one is this way of course there can be thousands of different ways but broadly one in which it is bowed out second in which it can be bowed in and third this is a straight line which one do you think is more likely so let us compare first something that is bowed in here is second input here is first input and this we have bowed out this is the second input this is the first input what is happening here let us look at it let us say we move this by one unit once we move it by one unit let us consider two situation in one in which the indifference curve is bowed in uh towards origin and here it is bowed out which is more likely let us see let us say this is 2 3 4 5 and you know here we have X2 so what we are doing in this case X1 is is increasing by one unit and because X1 is increasing by one unit X2 has to be decreased so in this is the case when this person has very little X1 and plenty of X2 so for for one unit of one unit more of X1 he is willing to give give a lot of X2 and I'm saying quantitatively okay the way I'm presenting is qualitative so for 2 to 3 let us say the jump is this much from 3 to 4 jump is smaller from 4 to 5 jump is even smaller so what we are saying when you have plenty of one item you value it less here because at two you have very little X1 so you value it a lot more if it is bowed out what it means that in this case you have plenty of X2 and this is X1 sorry this is X2 and you have very little of X1 let us say it is 1 2 3 4 5 so when you have very little of X1 your value for X1 in terms of X2 is also very little and you as you have more of X1 your value for X1 in terms of X2 is increasing which is more likely in our real life this is more likely whatever we have in less quantity our mind values it more and that's what the convexity of preference is what we say when we draw an indifference curve then we say we can say this is the Jone where you have higher higher utility bundle bundle what we can say a set you know you take any bundle X notot and you come up with the set that is preferred to X notot that would be the set that's the way we are indicating and that set has to be convex that's what convexity of preference mean but by the way I gave you a mathematical way to talk about it we can also talk about it in economic way so let us look at it what's happening here if this is the shape it's bowed in 2 1 we take a bundle here let us call this bundle X we take a bundle here Y and if we draw a line that passes through X and Y let us say this is the straight line my poor drawing uh what we mean that all the bundle on the line connecting X and Y would be preferred over X and Y because why because this is on this side here the utility is higher more is better that's why utility is increasing in this direction so how can we represent any bundle that lies between X and Y we can say Lambda x + 1 - Lambda Y where Lambda is between 0 and 1 what we mean to say that all these bundles would be at least as good as X will or because X and Y are giving the same same utility we have Lambda x + 1 - Lambda Y is also at least as good as y this we say that weak convexity we can have even stronger requirement we can say that Lambda x + 1us Lambda Y is strictly preferred to X and is strictly prefer to Y for all Lambda between 0 and 1 and notice here I have included 0 and one and here I'm not including 0 and one why because if I had included let us say one it would have been X preferred to X which would have been the wrong statement and this is a strict convexity we can give uh an economic interpretation to it also what is that economic interpretation economic interpretation is that balanced balanced bundle is preferred preferred over extreme bundles bundles on the same indifference curve oh sorry not on the same indifference curve just a balanced bundle is preferred over uh extreme bundles so I hope convexity is clear to you let me again restate what I am trying to say that rationality assumptions are absolutely required if an individual who is decision maker is not rational we cannot talk about that person in meaningful manner continuity helps us translate into utility function which makes our life easier but even if it is not continuous we can still talk about that person so once we have rationality and continuity we get continuous utility function to represent that person's preferences all these psychological assumption this is what we encounter more and more in real life but they can be satisfied they may not be satisfied we should not worry if they are not satisfied for example let us look at it uh you know um it's very much possible that there is a person whose indifference curve is like this you know we have we can say this is X2 this is X1 and as this person moves in this direction he gets happier and happier and therefore it is not convex because we take a bundle here we B another bundle here we draw the straight line and all these bundles are not more preferred to these bundles and therefore convexity is not satisfied one can say that uh um you know you like curd and you like chocolate separately but mixer is mixed is not good and so that a balance bundle will be less preferred you would rather like to have more of a chocolate or more of a curd but not a balanced bundle so that's also possible in life so we should allow for that utility function is not convex okay so that is the example I wanted to give you we can now talk about several utility function like we did in case of isoquant one we can have just let us say ax1 + bx2 you know is equal to K that is the utility function here goods are perfect substitute when there is a matter of confusion many people get confused used when we say they are perfect substitute they want the exchange ratio to be one that is not the requirement for the perfect substitute in other word Mr s may not necessarily be equal to one as long as mrss is constant mrss is not changing we will say goods are perfect substitute and goods are perfect complement in this particular case when we have this is the shape of utility function and this we already know how it is represented minimum of a X1 comma bx2 that's the way we can represent I already give you the example of fish and coconut which had a satiation point just very quickly we can look at that example and say only one point that you know here the monotonicity is V viated but local non satiation is satisfied except at the point that gives this person maximum utility and there it gets violated but everywhere else local nons satiation is satisfied so this is it on preferences in the next week we will do things mathematically we will get into the demand function and we will make progress in a practical manner this is this week's defin descript ion has been very theoretical in nature but theory is also important for us to understand all these things so see you next week thank you